A) \[\frac{5\sqrt{3}}{3}(\hat{i}+\hat{j}+\hat{k})\]
B) \[\frac{5\sqrt{3}}{3}(\hat{i}+\hat{j}-\hat{k})\]
C) \[\frac{5\sqrt{3}}{3}(\hat{i}-\hat{j}+\hat{k})\]
D) \[\frac{5\sqrt{3}}{3}(-\hat{i}+\hat{j}+\hat{k})\]
Correct Answer: A
Solution :
Let \[\overrightarrow{A}=(\hat{i}-2\hat{j}+\hat{k})\] \[\overrightarrow{B}=(2\hat{i}+\hat{j}-3\hat{k})\] Now, \[\overrightarrow{A}\times \overrightarrow{B}=\left| \begin{matrix} {\hat{i}} & {\hat{j}} & {\hat{k}} \\ 1 & -2 & 1 \\ 2 & 1 & -3 \\ \end{matrix} \right|\] \[=5\hat{i}+5\hat{j}+5\hat{k}\] \[|\overrightarrow{A}\times \overrightarrow{B}|=\sqrt{25+25+25}=5\sqrt{3}\] \[\therefore \]Unit vector along\[\overrightarrow{A}\times \overrightarrow{B}\]is \[\frac{5(\hat{i}+\hat{j}+\hat{k})}{5\sqrt{3}}\] \[\Rightarrow \]Required vector of magnitude 5 is \[\frac{5\sqrt{3}}{3}(\hat{i}+\hat{j}+\hat{k})\]
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